Three-Antenna Two-Dimensional Imaging Correlation Radiometer: Concept and Preliminary Results A. Camps, A. Sumpsi Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya Campus Nord, D4-016, E-08034 Barcelona, Spain Tel. +34+934016085, Fax. +34+934017232, e-mail: camps@tsc.upc.es Abstract This paper presents the concept and preliminary experimental results of a two-dimensional imaging correlation radiometer formed by three antennas located at the vertices of an equilateral triangle. This radiometer concept is based on the Doppler- radiometer described in [1]. Due to the limited data record, in the experimental results presented, Doppler focusing was not possible, and only delay focusing was used. The radiometer antennas form very long baselines and, since the transit time to travel to the antenna positions is larger than the correlation time, signal decorrelates. The instrumental delays inserted before the computation of the signals’ cross-correlations are then used to focus a particular direction, and correlating at different time lags an image is formed. I. BASIC PRINCIPLES The antenna array configuration is defined by the antenna positions: ( ) ( ) ( ) 1 2 1 3 1 2 0,0 , ,0 , 2, R R D R D D = = = . Three baselines, named 1-2, 2-3 and 3-1, can be formed from the corresponding antenna pairs: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 12 2 1 0 1 0 23 23 3 2 0 1 2 0 31 31 1 3 0 1 2 0 , ( ) ,0 ,0 , ˆ , ( ) 2, 2, , ˆ , ( ) 2, 2, . (1) ˆ u v R R D u u v R R D D u v u v R R D D u v λ λ λ λ λ λ = - = = = - =- =- = - = - = - where 0 0 cf λ = is the electromagnetic wavelength at the center frequency 0 f . System’s output ( ) 123 2 3 , V τ τ is formed by the intensity correlation of the signals ( ) 1 1 b t τ - , ( ) 2 2 b t τ - and ( ) 3 3 b t τ - collected by the three antennas, delayed with instrumental delays 1 0 τ = , 2 0 u f τ ξ = and ( ) 3 0 2 u v f τ ξ η = + are inserted to focus the signals coming from the direction ( ) ( ) , sin cos ,sin sin ξ η θ φ θ φ = : ( ) () () ( ) ( ) ( ) ( ) ( ) ( ) ( ) 123 2 3 * * * 1 1 1 2 2 2 2 2 3 3 3 3 3 , . (2) V b tb t I b t b t I b t b t I τ τ τ τ τ τ = = - - - - - - - where () () () 2 * n n n n I b tb t b t = = is the power of () n b t . Expanding eqn. (2) using of the circular joint Gaussian random variables Theorem [Goodman, 1985, p. 44], and assuming that 1 2 3 I I I = = , eqn. 2 reduces to: ( ) [ ] 123 2 3 12 23 31 , 2 Re V VVV τ τ = , (3) where ( ) ( ) * lm l l m m V b t b t τ τ = - - is the amplitude complex cross-correlation between the signals collected by the element pair l and m, with l, m=1,2,3. In synthetic aperture radiometry V lm is called a sample of the visibility function. For a point source located at ( ) ', ' ξ η with brightness temperature 0 T ( ( ) ( ) 0 , · ', ' B T T ξ η δ ξ ξ η η = - - ), V lm is given by the Van Cittert-Zernike Theorem [2, 3, 4, 5]: ( ) ( ) ( ) ( ) * * 0 2 2 0 0 0 1 2 1 ', ' ', ' 1 ' ' ' ' ' ' exp 2 , (4) lm l l m m l m lm lm lm lm lm l m l m m l n n V b t b t T F F u v u v r j f f f τ τ ξη ξη ξ η ξ η ξ η τ τ π τ τ = - - = ΩΩ - - + + - - - - where Ω l and F nl (ξ,η) are the antenna equivalent solid angle and normalized radiation voltage pattern of element l, 2 2 1 ξ η ′ ′ - - is the obliquity factor (cosine of the angle from the boresight direction), and () ( ) ( ) ( ) * 0 0 1 exp 2 lm l m m l n n r t BB H f H f j f t df π ∞ = ⋅ - is the fringe washing function that accounts for spatial decorrelation effects, and depends on the receiver l and m normalized frequency responses ( ) l n H f and ( ) m n H f . If all receivers have the same normalized antenna radiation voltage pattern ( ( ) , n F ξ η ), equivalent solid angle (Ω), and normalized frequency response, modeled by a Gaussian filter ( ) ( ) ( ) ( ) 2 2 0 exp 2 n H f f f B π = - - centered at 0 f f = with equivalent noise bandwidth B, then () lm r τ becomes ( ) ( ) 2 2 exp rt Bt π = - , and system’s output for a point source the impulse response is given by: ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 2 2 0 123 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 0 2 2 1 , 2 , exp 1 ·exp exp 2 2 1 3 2 , exp 2 2 1 · n n T V F Wu u u W v W v T F W u v τ τ ξη π ξ ξ ξ η ξ ξ ξ ξ π η η π η η ξη π ξ ξ η η ξ η ′ ′ ′ = - - Ω ′ ′ - - ′ ′ - - ′ ′ - - + - - + - = ′ ′ ′ ′ = - - + - Ω ′ ′ - - , (5) where 0 ˆ W Bf = is the relative bandwidth. If 32 v u = ⋅ , the array is an equilateral triangle with side u, and the impulse response ( 0 1 T = ) becomes a circularly-symmetric two-dimensional Gaussian function with 1/e widths ( ) 1 Wu ξ η σ σ π = = , or half-power beamwidths 3dB 3dB 2 2 ln 2 · 1.3 · Wu Wu ξ η π - - Δ =Δ = ≈ . 0-7803-7929-2/03/$17.00 (C) 2003 IEEE 2152